We would like to know in the cases 1a, 3a and 4a in the list, which one of the eigenvalues has the negative sign in the sign characteristic, and in cases 1b, 3b, 4b and 4c what the sign corresponding to the Jordan block of size two is. In order to answer these questions we first recall one of the descriptions of the sign characteristic, see Section 5.1 in [1]}. First note that for every real \(\lambda \) , the matrix \(\lambda H-HA\)  is an \(n\times n\)  Hermitian matrix, and hence has \(n\)  real eigenvalues, which are denoted by
\(\nu _1(\lambda ), \ldots , \nu _n(\lambda )\) . It turns out that these can be chosen to be analytic functions of the real variable \(\lambda \) , and this will be done for now. Let \(\lambda _1, \ldots , \lambda _r\)  be the real eigenvalues of \(A\) , and write for every \(i=1, \ldots , r\)  and \(j=1, \ldots , n\)  the function \(\nu _j(\lambda )\)  as
\(\nu _j(\lambda )=(\lambda -\lambda _i)^{m_{ij}}\rho _{ij}(\lambda ),\) 
